Per dire is a common expression. It has different variations, with different sfumature, but this is perhaps it's most synthetic variant. It's as if the speaker were saying per esempio (for example), or "let's say..." In both cases, it's practically a stand-alone expression that gets inserted in a sentence with a comma or an ellipsis.
Per dire, io prendo la pastiera napoletana,
Let's say I take the Neapolitan Pastiera:
all'interno c'è il grano, simbolo di ricchezza.
inside there's wheat, symbol of wealth.
Captions 33-34, Susanna Cutini - Dolci delle tradizioni di PasquaPlay Caption
Literally, per dire means "in order to say." Perhaps the best way to think of it is "for the sake of argument." We might even simply use "say," as in "suppose."
Let's take, say, a Neapolitan pastiera...
Suppose I take the Neapolitan pastiera...
We're primarily talking about a hypothetical example, which may or may not actually be a true-life example. Susanna's example about her grandfather happened to be true, but she was using it as an example.
Perché a Pasqua lui doveva avere lo zafferano
Because at Easter, he had to have saffron
per fare le panine pasquali. Per dire...
to make the Easter breads. Just as an example...
Captions 82-83, Susanna Cutini - Dolci delle tradizioni di PasquaPlay Caption